- For which values of $\alpha$ these topological spaces $X_{\alpha}$ are homotopy-equivalent to topological space $X_{0.5}$?
- For those spaces which are homotopy-equivalent to $X_{0.5}$ provide a homotopy equivalence.

## Homotopy equivalence of closed curves

- Grigorios Kostakos
- Founder
**Posts:**461**Joined:**Mon Nov 09, 2015 1:36 am**Location:**Ioannina, Greece

### Homotopy equivalence of closed curves

Consider the family \[C_{\alpha}:\big({2(\alpha-\cos{t})\,\cos{t},\,2(\alpha-\cos{t})\,\sin{t}}\big)\,,\;t\in[0,2\pi]\,,\;\alpha\in\mathbb{R},\] of closed parametric curves. Let $X_{\alpha}$ is the image set of $C_{\alpha}$ equipped with the induced topology of $\mathbb{R}^2$.

Grigorios Kostakos

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